--- a/src/ZF/OrderArith.thy Sun Mar 04 23:20:43 2012 +0100
+++ b/src/ZF/OrderArith.thy Tue Mar 06 15:15:49 2012 +0000
@@ -12,22 +12,22 @@
radd :: "[i,i,i,i]=>i" where
"radd(A,r,B,s) ==
{z: (A+B) * (A+B).
- (EX x y. z = <Inl(x), Inr(y)>) |
- (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r) |
- (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
+ (\<exists>x y. z = <Inl(x), Inr(y)>) |
+ (\<exists>x' x. z = <Inl(x'), Inl(x)> & <x',x>:r) |
+ (\<exists>y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
definition
(*lexicographic product of two relations; underlies ordinal multiplication*)
rmult :: "[i,i,i,i]=>i" where
"rmult(A,r,B,s) ==
{z: (A*B) * (A*B).
- EX x' y' x y. z = <<x',y'>, <x,y>> &
+ \<exists>x' y' x y. z = <<x',y'>, <x,y>> &
(<x',x>: r | (x'=x & <y',y>: s))}"
definition
(*inverse image of a relation*)
rvimage :: "[i,i,i]=>i" where
- "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
+ "rvimage(A,f,r) == {z: A*A. \<exists>x y. z = <x,y> & <f`x,f`y>: r}"
definition
measure :: "[i, i\<Rightarrow>i] \<Rightarrow> i" where
@@ -39,19 +39,19 @@
subsubsection{*Rewrite rules. Can be used to obtain introduction rules*}
lemma radd_Inl_Inr_iff [iff]:
- "<Inl(a), Inr(b)> : radd(A,r,B,s) <-> a:A & b:B"
+ "<Inl(a), Inr(b)> \<in> radd(A,r,B,s) <-> a:A & b:B"
by (unfold radd_def, blast)
lemma radd_Inl_iff [iff]:
- "<Inl(a'), Inl(a)> : radd(A,r,B,s) <-> a':A & a:A & <a',a>:r"
+ "<Inl(a'), Inl(a)> \<in> radd(A,r,B,s) <-> a':A & a:A & <a',a>:r"
by (unfold radd_def, blast)
lemma radd_Inr_iff [iff]:
- "<Inr(b'), Inr(b)> : radd(A,r,B,s) <-> b':B & b:B & <b',b>:s"
+ "<Inr(b'), Inr(b)> \<in> radd(A,r,B,s) <-> b':B & b:B & <b',b>:s"
by (unfold radd_def, blast)
lemma radd_Inr_Inl_iff [simp]:
- "<Inr(b), Inl(a)> : radd(A,r,B,s) <-> False"
+ "<Inr(b), Inl(a)> \<in> radd(A,r,B,s) <-> False"
by (unfold radd_def, blast)
declare radd_Inr_Inl_iff [THEN iffD1, dest!]
@@ -59,7 +59,7 @@
subsubsection{*Elimination Rule*}
lemma raddE:
- "[| <p',p> : radd(A,r,B,s);
+ "[| <p',p> \<in> radd(A,r,B,s);
!!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q;
!!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q;
!!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q
@@ -68,7 +68,7 @@
subsubsection{*Type checking*}
-lemma radd_type: "radd(A,r,B,s) <= (A+B) * (A+B)"
+lemma radd_type: "radd(A,r,B,s) \<subseteq> (A+B) * (A+B)"
apply (unfold radd_def)
apply (rule Collect_subset)
done
@@ -86,10 +86,10 @@
lemma wf_on_radd: "[| wf[A](r); wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))"
apply (rule wf_onI2)
-apply (subgoal_tac "ALL x:A. Inl (x) : Ba")
+apply (subgoal_tac "\<forall>x\<in>A. Inl (x) \<in> Ba")
--{*Proving the lemma, which is needed twice!*}
prefer 2
- apply (erule_tac V = "y : A + B" in thin_rl)
+ apply (erule_tac V = "y \<in> A + B" in thin_rl)
apply (rule_tac ballI)
apply (erule_tac r = r and a = x in wf_on_induct, assumption)
apply blast
@@ -116,7 +116,7 @@
lemma sum_bij:
"[| f: bij(A,C); g: bij(B,D) |]
- ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : bij(A+B, C+D)"
+ ==> (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) \<in> bij(A+B, C+D)"
apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))"
in lam_bijective)
apply (typecheck add: bij_is_inj inj_is_fun)
@@ -125,8 +125,8 @@
lemma sum_ord_iso_cong:
"[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |] ==>
- (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))
- : ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
+ (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))
+ \<in> ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
apply (unfold ord_iso_def)
apply (safe intro!: sum_bij)
(*Do the beta-reductions now*)
@@ -134,9 +134,9 @@
done
(*Could we prove an ord_iso result? Perhaps
- ord_iso(A+B, radd(A,r,B,s), A Un B, r Un s) *)
-lemma sum_disjoint_bij: "A Int B = 0 ==>
- (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)"
+ ord_iso(A+B, radd(A,r,B,s), A \<union> B, r \<union> s) *)
+lemma sum_disjoint_bij: "A \<inter> B = 0 ==>
+ (\<lambda>z\<in>A+B. case(%x. x, %y. y, z)) \<in> bij(A+B, A \<union> B)"
apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective)
apply auto
done
@@ -144,16 +144,16 @@
subsubsection{*Associativity*}
lemma sum_assoc_bij:
- "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
- : bij((A+B)+C, A+(B+C))"
+ "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
+ \<in> bij((A+B)+C, A+(B+C))"
apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))"
in lam_bijective)
apply auto
done
lemma sum_assoc_ord_iso:
- "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
- : ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),
+ "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
+ \<in> ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),
A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))"
by (rule sum_assoc_bij [THEN ord_isoI], auto)
@@ -163,14 +163,14 @@
subsubsection{*Rewrite rule. Can be used to obtain introduction rules*}
lemma rmult_iff [iff]:
- "<<a',b'>, <a,b>> : rmult(A,r,B,s) <->
+ "<<a',b'>, <a,b>> \<in> rmult(A,r,B,s) <->
(<a',a>: r & a':A & a:A & b': B & b: B) |
(<b',b>: s & a'=a & a:A & b': B & b: B)"
by (unfold rmult_def, blast)
lemma rmultE:
- "[| <<a',b'>, <a,b>> : rmult(A,r,B,s);
+ "[| <<a',b'>, <a,b>> \<in> rmult(A,r,B,s);
[| <a',a>: r; a':A; a:A; b':B; b:B |] ==> Q;
[| <b',b>: s; a:A; a'=a; b':B; b:B |] ==> Q
|] ==> Q"
@@ -178,7 +178,7 @@
subsubsection{*Type checking*}
-lemma rmult_type: "rmult(A,r,B,s) <= (A*B) * (A*B)"
+lemma rmult_type: "rmult(A,r,B,s) \<subseteq> (A*B) * (A*B)"
by (unfold rmult_def, rule Collect_subset)
lemmas field_rmult = rmult_type [THEN field_rel_subset]
@@ -195,7 +195,7 @@
apply (rule wf_onI2)
apply (erule SigmaE)
apply (erule ssubst)
-apply (subgoal_tac "ALL b:B. <x,b>: Ba", blast)
+apply (subgoal_tac "\<forall>b\<in>B. <x,b>: Ba", blast)
apply (erule_tac a = x in wf_on_induct, assumption)
apply (rule ballI)
apply (erule_tac a = b in wf_on_induct, assumption)
@@ -221,7 +221,7 @@
lemma prod_bij:
"[| f: bij(A,C); g: bij(B,D) |]
- ==> (lam <x,y>:A*B. <f`x, g`y>) : bij(A*B, C*D)"
+ ==> (lam <x,y>:A*B. <f`x, g`y>) \<in> bij(A*B, C*D)"
apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>"
in lam_bijective)
apply (typecheck add: bij_is_inj inj_is_fun)
@@ -231,20 +231,20 @@
lemma prod_ord_iso_cong:
"[| f: ord_iso(A,r,A',r'); g: ord_iso(B,s,B',s') |]
==> (lam <x,y>:A*B. <f`x, g`y>)
- : ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
+ \<in> ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
apply (unfold ord_iso_def)
apply (safe intro!: prod_bij)
apply (simp_all add: bij_is_fun [THEN apply_type])
apply (blast intro: bij_is_inj [THEN inj_apply_equality])
done
-lemma singleton_prod_bij: "(lam z:A. <x,z>) : bij(A, {x}*A)"
+lemma singleton_prod_bij: "(\<lambda>z\<in>A. <x,z>) \<in> bij(A, {x}*A)"
by (rule_tac d = snd in lam_bijective, auto)
(*Used??*)
lemma singleton_prod_ord_iso:
"well_ord({x},xr) ==>
- (lam z:A. <x,z>) : ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
+ (\<lambda>z\<in>A. <x,z>) \<in> ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
apply (rule singleton_prod_bij [THEN ord_isoI])
apply (simp (no_asm_simp))
apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl])
@@ -253,9 +253,9 @@
(*Here we build a complicated function term, then simplify it using
case_cong, id_conv, comp_lam, case_case.*)
lemma prod_sum_singleton_bij:
- "a~:C ==>
- (lam x:C*B + D. case(%x. x, %y.<a,y>, x))
- : bij(C*B + D, C*B Un {a}*D)"
+ "a\<notin>C ==>
+ (\<lambda>x\<in>C*B + D. case(%x. x, %y.<a,y>, x))
+ \<in> bij(C*B + D, C*B \<union> {a}*D)"
apply (rule subst_elem)
apply (rule id_bij [THEN sum_bij, THEN comp_bij])
apply (rule singleton_prod_bij)
@@ -268,10 +268,10 @@
lemma prod_sum_singleton_ord_iso:
"[| a:A; well_ord(A,r) |] ==>
- (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))
- : ord_iso(pred(A,a,r)*B + pred(B,b,s),
+ (\<lambda>x\<in>pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))
+ \<in> ord_iso(pred(A,a,r)*B + pred(B,b,s),
radd(A*B, rmult(A,r,B,s), B, s),
- pred(A,a,r)*B Un {a}*pred(B,b,s), rmult(A,r,B,s))"
+ pred(A,a,r)*B \<union> {a}*pred(B,b,s), rmult(A,r,B,s))"
apply (rule prod_sum_singleton_bij [THEN ord_isoI])
apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE)
@@ -281,25 +281,25 @@
lemma sum_prod_distrib_bij:
"(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
- : bij((A+B)*C, (A*C)+(B*C))"
+ \<in> bij((A+B)*C, (A*C)+(B*C))"
by (rule_tac d = "case (%<x,y>.<Inl (x),y>, %<x,y>.<Inr (x),y>) "
in lam_bijective, auto)
lemma sum_prod_distrib_ord_iso:
"(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
- : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),
+ \<in> ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),
(A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
by (rule sum_prod_distrib_bij [THEN ord_isoI], auto)
subsubsection{*Associativity*}
lemma prod_assoc_bij:
- "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) : bij((A*B)*C, A*(B*C))"
+ "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) \<in> bij((A*B)*C, A*(B*C))"
by (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective, auto)
lemma prod_assoc_ord_iso:
"(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)
- : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),
+ \<in> ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),
A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
by (rule prod_assoc_bij [THEN ord_isoI], auto)
@@ -307,12 +307,12 @@
subsubsection{*Rewrite rule*}
-lemma rvimage_iff: "<a,b> : rvimage(A,f,r) <-> <f`a,f`b>: r & a:A & b:A"
+lemma rvimage_iff: "<a,b> \<in> rvimage(A,f,r) <-> <f`a,f`b>: r & a:A & b:A"
by (unfold rvimage_def, blast)
subsubsection{*Type checking*}
-lemma rvimage_type: "rvimage(A,f,r) <= A*A"
+lemma rvimage_type: "rvimage(A,f,r) \<subseteq> A*A"
by (unfold rvimage_def, rule Collect_subset)
lemmas field_rvimage = rvimage_type [THEN field_rel_subset]
@@ -361,7 +361,7 @@
lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
apply clarify
-apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }")
+apply (subgoal_tac "\<exists>w. w \<in> {w: {f`x. x:Q}. \<exists>x. x: Q & (f`x = w) }")
apply (erule allE)
apply (erule impE)
apply assumption
@@ -373,7 +373,7 @@
@{text wf_rvimage} gives @{prop "wf(r) ==> wf[C](rvimage(A,f,r))"}*}
lemma wf_on_rvimage: "[| f: A->B; wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
apply (rule wf_onI2)
-apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba")
+apply (subgoal_tac "\<forall>z\<in>A. f`z=f`y \<longrightarrow> z: Ba")
apply blast
apply (erule_tac a = "f`y" in wf_on_induct)
apply (blast intro!: apply_funtype)
@@ -396,7 +396,7 @@
done
lemma ord_iso_rvimage_eq:
- "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A"
+ "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r \<inter> A*A"
by (unfold ord_iso_def rvimage_def, blast)
@@ -440,7 +440,7 @@
lemma wf_imp_subset_rvimage:
- "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
+ "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r \<subseteq> rvimage(A, f, Memrel(i))"
apply (rule_tac x="wftype(r)" in exI)
apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
apply (simp add: Ord_wftype, clarify)
@@ -450,25 +450,25 @@
done
theorem wf_iff_subset_rvimage:
- "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
+ "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r \<subseteq> rvimage(A, f, Memrel(i)))"
by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
intro: wf_rvimage_Ord [THEN wf_subset])
subsection{*Other Results*}
-lemma wf_times: "A Int B = 0 ==> wf(A*B)"
+lemma wf_times: "A \<inter> B = 0 ==> wf(A*B)"
by (simp add: wf_def, blast)
text{*Could also be used to prove @{text wf_radd}*}
lemma wf_Un:
- "[| range(r) Int domain(s) = 0; wf(r); wf(s) |] ==> wf(r Un s)"
+ "[| range(r) \<inter> domain(s) = 0; wf(r); wf(s) |] ==> wf(r \<union> s)"
apply (simp add: wf_def, clarify)
apply (rule equalityI)
prefer 2 apply blast
apply clarify
apply (drule_tac x=Z in spec)
-apply (drule_tac x="Z Int domain(s)" in spec)
+apply (drule_tac x="Z \<inter> domain(s)" in spec)
apply simp
apply (blast intro: elim: equalityE)
done
@@ -496,7 +496,7 @@
lemma wf_measure [iff]: "wf(measure(A,f))"
by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
-lemma measure_iff [iff]: "<x,y> : measure(A,f) <-> x:A & y:A & f(x)<f(y)"
+lemma measure_iff [iff]: "<x,y> \<in> measure(A,f) <-> x:A & y:A & f(x)<f(y)"
by (simp (no_asm) add: measure_def)
lemma linear_measure:
@@ -521,7 +521,7 @@
apply (blast intro: linear_measure Ordf inj)
done
-lemma measure_type: "measure(A,f) <= A*A"
+lemma measure_type: "measure(A,f) \<subseteq> A*A"
by (auto simp add: measure_def)
subsubsection{*Well-foundedness of Unions*}
@@ -549,7 +549,7 @@
lemma Pow_sum_bij:
"(\<lambda>Z \<in> Pow(A+B). <{x \<in> A. Inl(x) \<in> Z}, {y \<in> B. Inr(y) \<in> Z}>)
\<in> bij(Pow(A+B), Pow(A)*Pow(B))"
-apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} Un {Inr (y). y \<in> Y}"
+apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} \<union> {Inr (y). y \<in> Y}"
in lam_bijective)
apply force+
done